Problem:
A long piece of paper 5 cm wide is made into a roll for cash registers by wrapping it 600 times around a cardboard tube of diameter 2 cm, forming a roll 10 cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms 600 concentric circles with diameters evenly spaced from 2 cm to 10 cm.)
Answer Choices:
A. 36Ï€
B. 45Ï€
C. 60Ï€
D. 72Ï€
E. 90Ï€
Solution:
Let d1​,d2​,…,d600​ be the diameters of the concentric circles in the model. The d 's form an arithmetic sequence with d1​=2 cm and d600​=10 cm. If L is the total length, then
L​=πd1​+πd2​+⋯+πd600​=π(d1​+d2​+⋯+d600​)=π600(2d1​+d600​​)=π600(212​)cm=36π meters. ​
OR
Let L be the length of the tape in cm. The thickness of paper on the roll is (10−2)/2=4 cm. Therefore, the thickness of the tape is 4/600=1/150 cm. We may assume that unfolding the paper and laying it out flat has negligible effect on its cross-sectional area. Therefore, we may equate the cross-sectional area of the laid out paper, which is L/150 cm2, to the cross-sectional area while it is on the roll, which is
π52−π12=24π cm2
Solving for L gives L=3600Ï€ cm=36Ï€ meters.